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Explicit phase diagram for a one-dimensional blister model
Ghada Chmaycem, Mustapha Jazar, Régis Monneau
To cite this version:
Ghada Chmaycem, Mustapha Jazar, Régis Monneau. Explicit phase diagram for a one-dimensional blister model. 2014. �hal-00947515�
Explicit phase diagram
for a one-dimensional blister model
G. Chmaycemb,∗, M. Jazar ∗and R. Monneaub
February 16, 2014
Abstract
We consider a thin film bonded to a substrate. The film acquires a residual stress upon cooling because of the mismatch of thermal expansion coefficient between the film and the substrate. The film tends to lift off the substrate when this residual stress is compressive and large enough. In this work, this phenomenon is described by a simplified one-dimensional variational model. We minimize an energy and study its global minimizers. Our problem depends on three parameters: the length of the film, its elasticity and a thermal parameter.
Our main result consists in describing a phase diagram depending on those parameters in order to identify three types of global minimizers: a blister, a fully delaminated blister and a trivial solution (without any delamination). Moreover, we prove various qualitative results on the shape of the blisters and identify the smallest blister which may appear.
Keywords: blister, thin film, fracture, delamination, buckling, Föppel-von Kármán, variational model, classification of global minimizers, phase diagram, nonlinear elasticity, obstacle problem, non interpenetration condition.
1 Introduction
1.1 Physical motivation
The thin films are often obtained by evaporation on a substrate. When the coefficient of thermal expansion of the substrate is higher than that of the film, cooling to ambient tem- perature leads to a compressive residual stress in the film. If compression is sufficient, the film tends to buckle, separating from the substrate. It is said that the film delaminates (see Figure 1).
∗LaMA-Liban, Lebanese University, P.O. Box 37 Tripoli, Lebanon. E-mail: ghada.chmaycem@gmail.com (G. Chmaycem), mjazar@laser-lb.org (M. Jazar).
b Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. E-mail: chemaycg@cermics.enpc.fr (G. Chmaycem), monneau@cermics.enpc.fr (R. Monneau)
An oversimplified one dimensional model which describes this phenomena is given by the minimization of the following energy (of Föppel-von Kármán type)
E(ζ1, ζ2) :=
Z
Ω
γ1{ζ2>0}+ 4α
ζ1′ + 1 2(ζ2′)2
2
+ 4α
3 ζ2′′2 −2θζ2′2 with γ = 1, (1.1) with
Ω =R/LZ= [−L/2, L/2[per,
and where L is the length of the film, α > 0 represents its elasticity coefficient and θ > 0 is the thermal parameter. Here the parameter γ measures the cost of delamination and is similar to the formulation of fracture with Griffith criterion (see for instance Francfort, Marigo [6], Griffith [7], Larsen [9]). For γ = 0, this model was formally derived from 3D elasticity in the asymptotics of thin films in [5] by El Doussouki and the last author, see also [10]. For simplicity, we normalize this parameter γ to be equal to 1in the whole paper (this normalization can always be absorbed in a redefinition of E, α and θ by rescaling). The quantity ζ2(x) denotes the vertical displacement and is assumed to be nonnegative (the film is above the substrate) and ζ1(x) is the horizontal one with x ∈ Ω, where the periodicity is assumed to simplify the analysis (see also Remark 1.4 for other boundary conditions describing a clamped film). We introduce the following space
Y :=H1(Ω)× {ζ2 ∈H2(Ω), ζ2 ≥0}. (1.2) The solution of our model is given by solving the following problem
(ζ1min,ζ2)∈Y E(ζ1, ζ2). (1.3) Definition 1.1 (Blisters)
We call a "blister" any global minimizer of the energy E defined in (1.1) which is non trivial i.e. (ζ1, ζ2)6≡(0,0).
This paper elaborates the delamination of compressed thin films. Under appropriate condi- tions, blisters may appear. We give a complete description of global minimizers in terms of the parameters of the problem.
Figure 1: Different types of solutions of problem (1.3)
1.2 Main results
Theorem 1.2 (Existence of global minimizers)
There exists a (global) minimizer ζ = (ζ1, ζ2)∈Y of the energy E introduced in (1.1).
In order to study minimizers ofE, it is useful to consider the following auxiliary minimizing problem
minX∈Df(X), (1.4)
where
f(X) :=
(θ−X)−1/2−LX2 if 0< X < θ;
0 if X = 0;
(1.5) with rescaled versions of the thermal parameterθ and of the length L
θ := θ
α, and L:= 1 2π
r3
2αL, (1.6)
where α >0is from now on fixed in the model and D is the interval given by D :=h
0,θe+i
, with θe+ = maxn θ,e 0o
and θe:=θ− α2
L2. (1.7)
Indeed the following theorem shows that the minimizing problem (1.3) is equivalent to the study of the auxilary problem (1.4).
Theorem 1.3 (Description of global minimizers of E) i) (Implication)
For any global minimizer ζ = (ζ1, ζ2) of the energy E, there exists at least a minimizer K ∈ D of problem (1.4) such that the following holds: there existsT ∈[0, L] such that (up to
addition of constants and translation of (ζ1, ζ2)), this minimizer ζ can be written as follows
ζ1(x) =
K
2
x+ L 2
in [−L/2,−T /2];
KL
8π sin(2βx) + K 2
1− L
T
x in (−T /2, T /2);
K 2
x− L
2
in [T /2, L/2],
ζ2(x) =
A(cos(βx) + 1) in [−T /2, T /2];
0 elsewhere,
(1.8)
where β, A and T are given by
β :=
s
3(θ−αK)
2α ; A:=
s KL
πβ ; T :=
2π
β if K >0;
0 if K = 0.
(1.9)
More generally, for any K ∈ D and any functions (ζ1, ζ2) given in (1.8)-(1.9), we have E(ζ1, ζ2) = 2π
r2
3f(K), (1.10)
and for K ∈ D
T =L ⇔ K =θ;e and
T < L ⇔ K <θ.e
(1.11)
ii) (Reciprocal)
If K ∈ D is a minimizer of problem (1.4), then the function ζ = (ζ1, ζ2)given in (1.8)-(1.9) is a global minimizer of E on Y.
Notice that θ−αK > 0 because K ∈ D. Moreover, when K = 0 then A = T = 0 which implies that ζ1 =ζ2 = 0. Thus with our definition,T can be interpreted as the length of the support of ζ2. Theorem 1.3 identifies three types of global minimizers. For K = 0, we get the trivial solution (Figure 1, (a)). For K ∈ (0,θ), thene 0< T < L and we get the blister solution (Figure 1, (b)). Finally, for K = θ, thene T = L and we get the fully delaminated blister (Figure 1, (c)). We still use the name "blister" for the mathematical solution even if physically the film is completely delaminated. Note that our blister solution (Figure 1, (b)) can be roughly speaking seen as the cross section of blisters with the shape of fingers (see for instance experiments in Figure 8.1 in [12]).
Remark 1.4 (Clamped boundary conditions)
Recall that the periodic boundary conditions are included in the set Y defined in (1.2). We now introduce another set of functions satisfying clamped boundary conditions
Ye :=H01(−L/2, L/2)× {ζ2 ∈H02(−L/2, L/2), ζ2 ≥0}. Then
infYe
E ≥inf
Y E,
because any y∈Ye can be seen as an element ofY when it is extended by periodicity. More- over, any global minimizer of E on Y is given (up to addition of constants and translation of (ζ1, ζ2)) by the solution written in (1.8) which satisfies (ζ1, ζ2)∈Ye. Therefore,
infYe
E = inf
Y E,
and then in this paper we also solved the minimization problem of E on Ye.
To classify the solutions obtained in Theorem 1.3, we have to define the following functions in order to introduce some domains D0, D1 and D2 of parameters(θ, L). Figure 2 describes those domains (still for arbitrary fixed value α). We will show that trivial solutions corre- spond to D0, blister solutions to D1 and fully delaminated blister to D2. For this purpose, we introduce
θ∗ := 5
4α−1/2; (1.12)
L01(θ) := 55/2
16 θ−5/2 for 0< θ≤θ∗; (1.13)
L02(θ) := α5/4 (√
αθ−1)1/2 for θ≥θ∗ > α−1/2; (1.14) L12(θ) :=
2α3θ+ 2α2p
α(αθ2−1)1/2
for θ ≥θ∗ > α−1/2. (1.15) Definition 1.5 (Domains D0, D1 and D2)
Let us now introduce the following sets of (θ, L)∈(0,+∞)2:
D0 :=
(θ, L), L < L01(θ) if 0< θ≤θ∗ L < L02(θ) if θ > θ∗
; (1.16)
D1 :=
(θ, L), L > L01(θ) if 0< θ≤θ∗ L > L12(θ) if θ > θ∗
; (1.17)
D2 :={(θ, L), θ > θ∗ and L02(θ)< L < L12(θ)}. (1.18)
We denote by
Γ01 :={(θ, L),0< θ < θ∗ and L=L01(θ)}; (1.19) Γ02 :={(θ, L), θ > θ∗ and L=L02(θ)}; (1.20) Γ12 :={(θ, L), θ > θ∗ and L=L12(θ)}; (1.21)
P = (θ∗, L01(θ∗)). (1.22)
Remark 1.6 (A partition of the domains) We have the following disjoint decomposition
(0,+∞)2 =D0 ∪D1∪D2∪Γ21∪Γ01∪Γ02∪ {P}. Moreover, the following properties hold true
L′12(θ)>0 and L′02(θ)<0, for θ > θ∗; L′01(θ)<0, for 0< θ < θ∗; L∗ :=L12(θ∗) = L01(θ∗) = L02(θ∗);
(1.23)
where θ∗ is defined in (1.12).
The proof of Remark 1.6 is done by a simple computation.
Figure 2: Schematic phase diagram for parameters (θ, L)
Theorem 1.7 (Classification of global minimizers of E)
i) For (θ, L) ∈ D0, the unique global minimizer of the energy E introduced in (1.1) is the trivial solution (ζ1, ζ2) = (0,0).
ii) For (θ, L)∈D1∪D2∪Γ12, there is a unique blister ζ = (ζ1, ζ2)∈Y (see Definition 1.1) minimizing the energy E. Moreover, the component ζ2 has a support of length T which is defined in (1.9) and
T < L if (θ, L)∈D1; T =L if (θ, L)∈D2∪Γ12.
(1.24) iii) For (θ, L)∈Γ01∪Γ02∪ {P}, the energy E has exactly two global minimizers: the trivial solution ζ = (ζ1, ζ2) = (0,0) and a blister ζ = (ζ1, ζ2)∈Y given in (1.8) with
T < L if (θ, L)∈Γ01;
T =L if (θ, L)∈Γ02∪ {P}.
(1.25)
Proposition 1.8 (Blister’s properties in D1∪D2∪Γ12)
For (θ, L) ∈ D1 ∪D2∪Γ12, there exists a unique K ∈ D (depending on (θ, L)) minimizing problem (1.4). Recalling (1.6), we consider T and A given in (1.9).
i) Monotonicity
First,T andA are continuous in (θ, L) onD1∪D2∪Γ12 and satisfy the following properties
∂T
∂θ,∂T
∂L ≥0 and ∂A
∂θ,∂A
∂L ≥0.
In particular,
T =L on D2∪Γ12. ii) "Smallest" blister solutions
We have
(θ,L)∈Dinf 1
T =T∗ := 4π r2
3α1/4, (1.26)
and
(θ,L)∈Dinf 1
A=A∗ := 4
√3. (1.27)
Remark 1.9 (Prediction for the smallest blisters; not fully delaminated case) For any (θ, L) ∈ D1, we have a unique blister (ζ1, ζ2) ∈ Y minimizing the energy E. Ac- cording to Proposition 1.8, the second component ζ2 has a support of length T > T∗ with T < L. This shows thatT∗ can be interpreted as the infinimum of the width of blisters whose length support is strictly less than the length of the film. Similarly, we can also interpret the amplitude A∗ as the minimal amplitude of the blisters.
Remark 1.10 (Relatively small blisters for large L)
For (θ, L)∈D1, it is possible to check, as L tends to infinity, that T and A have a behavior like L1/3 and L2/3 respectively. In particular, for θ fixed and for large enough films, the size of the blisters is much smaller than the size of the film.
Remark 1.11 (Phase diagram for local minimizers)
For local minimizers (with small perturbation of the support with T < L), we expect to have similar phase diagram for pblisters (only local minimizers) of the same width T with now L replaced by L/p, L replaced by L/p and T ≤L/p (see Remark 2.2).
1.3 Brief review of the literature
Buckling delamination blisters have commonly been studied for a long time. In [8], Gioia and Ortiz give an overview of experiments, propose and study mathematically variational models of blisters, among other things motivated by the description of telephone-cord morphology.
See also [1] where such a telephone-cord instability is studied.
Experimentally and theoretically in [11], the authors study blisters which have the one dimensional symmetry. Their results seem coherent with ours, even if the problem and the modeling are not exactly the same. We also refer the reader to [3] and the references therein for recent developments on the analysis and modeling of blisters. In this nice work, the authors consider a Föppel-von Kármán model for the film with a special bonding energy with the substrate. For this variational model, they study several regimes for the energy. This is also interesting to mention the work [2] where the authors derive rigorously a variational similar model of thin films bonded to a substrate when the thickness of the film goes to zero.
Their limit energy contains in particular a bonding term which is similar to our term with γ in the energy (1.1).
1.4 Organization of the paper
The organization of the paper is as follows. In Section 2, we prove Theorems 1.2 and 1.3 i) on the existence and the description of global minimizers. Section 3 is dedicated to the detailed classification of global minimizers and their qualitative properties. There we prove Theorem 1.7, Theorem 1.3 ii) and Proposition 1.8. To this end, we divided this section into three parts. In the first one, we present some results which will be useful to prove Theorem 1.7. The second subsection is devoted to prove Theorem 1.7. We end up Section 3 by the proofs of Theorem 1.3 ii) and Proposition 1.8.
2 Proofs of Theorems 1.2 and 1.3 i)
This section is divided into two parts: the first one is devoted to prove Theorem 1.2 and the second is dedicated to the proof of Theorem 1.3 i).
2.1 Existence of global minimizers
Proof of Theorem 1.2
The proof of Theorem 1.2 is very classical. By considering a minimizing sequence ζk = (ζ1k, ζ2k)∈Y such thatE(ζk) −→
k→∞I = inf
ζ∈YE(ζ), and using Young’s inequality, we get 2θ
Z
Ω
(ζ2k′)2 = 4θ Z
Ω
ζ1k′ + 1 2(ζ2k′)2
≤ 2θ2L α + 2α
Z
Ω
ζ1k′+ 1 2(ζ2k′)2
2
. (2.1)
Thus we can bound the energy E(ζk). We will skip the steps of the proof since the result can be obtained in a classical way (see also [10] and [4]).
2.2 Description of global minimizers of E
We first start this subsection by the following lemma which will be used to prove Theorem 1.3 i).
Lemma 2.1 (Classification of solutions ζ2)
Let ζ2 ∈ {f ∈H2(Ω), f ≥0}. Consider the following ordinary differential equation
ζ2(4)+3(θ−αK)
2α ζ2′′= 0 on ω0 = (x0, y0);
ζ2 >0 on ω0;
ζ2 = 0 on ∂ω0;
(2.2)
where x0 < y0.
If θ−αK ≤0, then there is no solution of (2.2).
If θ−αK >0, then up to translate ζ2, we have
x0 =−T
2 and y0 = T 2;
ζ2(x) = A0(cos(βx) + 1) on ω0 = (x0, y0);
β = s
3(θ−αK)
2α = 2π
T ;
(2.3)
where A0 >0 is a constant.
Proof of Lemma 2.1
Since ζ2 ∈H2(R/LZ), then ζ2 ∈C1(R/LZ). Moreover ζ2 ≥0, which implies that
ζ2(x0) =ζ2′(x0) = 0 =ζ2(y0) =ζ2′(y0). (2.4) We can write ζ2 as ζ2(x) =ζ2S(x) +ζ2A(x) onω0 = (x0, y0), where ζ2S is the symmetric part of ζ2 and ζ2A is its anti-symmetric part. In particular ζ2S verifies the following conditions
ζ2S >0 on ω0 and ζ2S = (ζ2S)′ = 0 on∂ω0. (2.5) We skip the details of the proof which is a routine exercise.
Proof of Theorem 1.3 i)
Let(ζ1, ζ2)∈Y be a minimizer of E.
Step 1: Differentiating E with respect to ζ1
Differentiating E with respect toζ1 leads to the following Euler-Lagrange equation:
ζ1′ +1
2(ζ2′)2 ′
= 0 on Ω, i.e.
ζ1′ + 1
2(ζ2′)2 = K
2 where K = 1 L
Z
Ω
(ζ2′)2 ≥0. (2.6) Therefore the total energy becomes
E(ζ1, ζ2) =E(ζ2) :=
Z
Ω
1{ζ2>0}+α L
Z
Ω
(ζ2′)2 2
+4α 3
Z
Ω
(ζ2′′)2−2θ Z
Ω
(ζ2′)2.
IfK = 0, thenζ2 ≡0and thus ζ1 ≡const onΩ, and up to subtract a constant toζ1, we can assume that ζ1 ≡0.
If K >0, then ζ2 6≡0and we proceed as follows.
Step 2: Differentiating E with respect to ζ2
Differentiating E with respect toζ2, yields the following Euler-Lagrange equation ζ2(4)+3(θ−αK)
2α ζ2′′ = 0 on {ζ2 >0}. Up to add a constant to ζ2, we can assume that inf
Ω ζ2 = 0. Therefore there exists x0 ∈ Ω such that ζ2(x0) = 0. Up to translation, we choose x0 =−L/2. Then, we deduce that
{ζ2 >0}= ∪
i∈Jωi, for ωi = (xi, yi),
where J is a set at most countable and such that ωi ∩ωj =∅ for i 6= j. Applying Lemma 2.1 to each ωi, we conclude that θ−αK >0,
yi−xi =T = 2π
s 2α 3(θ−αK)
and up to translation, the solution ζ2 is given by (2.3) on each ωi with the amplitude A0
replaced by Ai. Now, we deduce that card(J) = p < +∞ with p ≥ 1 satisfying pT ≤ L.
Since the ωi are disjoint, we get KL=
Z
Ω
(ζ2′)2 = β2T 2
Xp
i=1
A2i. (2.7)
Hence using (2.3), we get
E(ζ2) =E(K, p) := 2pπ
s 2α
3(θ−αK) −αK2L (2.8)
with the conditions
1≤p≤ L
T and T = 2π
β . (2.9)
Then we minimize the energy with respect to pand we get the results.
Remark 2.2 (Local minimizers)
For local minimizers of E, we may have p blisters (all separated by any positive distance) with the same width T and with amplitude Ai satisfying (2.7). For p≥1 given, we can also optimize K in E(K, p)which should correspond to local minimizers of E (restricted to small perturbations of the support with T < L) with L replaced by L/p, L replaced by L/p and T ≤L/p.
3 Proofs of Theorem 1.7, Theorem 1.3 ii) and Proposition 1.8
Our aim is to prove Theorem 1.7, Theorem 1.3 ii) and Proposition 1.8. For this purpose, this section is divided into several parts. In the first one, we give some tools which will be useful to prove Theorem 1.7. The second subsection is dedicated to the proof of Theorem 1.7. Finally, we prove Theorem 1.3 ii) and Proposition 1.8 in the last subsection.
3.1 Preliminaries
First, we are interested in the following auxiliary minimization problem
minX∈Df(X) = f(K), (3.1)
where f and D are defined respectively in (1.5) and (1.7). Recall that for (θ, L) ∈(0,∞)2, we have that D ⊂
6= [0, θ). In order to determine the minimum of the function f on [0, θ), we have to introduce the quantity
Ld =Ld(θ) := 25 24
r5
3θ−5/2, for θ > 0. (3.2) Proposition 3.1 (Minimizing f on [0, θ) and its consequences)
We consider f defined in (1.5) and Ld introduced in (3.2).
i) If L≤Ld, then f is increasing on (0, θ) and argmin
D
f ={0}. In this case, we set artificially Xm := 2θ/5.
ii) If L > Ld, then there exist XM and Xm such that 0< XM <2θ/5< Xm < θ and
f′ ≥0 on (0, XM]∪[Xm, θ);
f′ <0 on (XM, Xm);
and
argmin
D
f ⊂ {0, Xm}. (3.3)
iii) Moreover for L > Ld, the function Xm =Xm(θ, L) is smooth and satisfies
∂θXm = 3/4(θ−Xm)−5/2
3/4(θ−Xm)−5/2−2L >1, (3.4) and
∂LXm = 2Xm
3/4(θ−Xm)−5/2−2L >0. (3.5) Proof of Proposition 3.1
Step 1: Proof of i) and ii) For0< X < θ, we have
f′(X) = g(X)−h(X) with g(X) := 1
2(θ−X)−3/2 and h(X) := 2LX.
We notice that g is strictly convex on [0, θ), with g(0) > 0 and g′(0) > 0. Therefore there exists a unique L = Ld > 0 such that the straight line y =h(X) is tangent from below to the graph y=g(X), at a point Xd>0. In particular, we have
g(Xd) = h(Xd);
g′(Xd) = h′(Xd).
The unique solution of this system is Xd = 2θ/5 and the value ofLd given in (3.2). Using the strict convexity of g (and the fact that g(θ−) = +∞), we deduce the variations of f in cases i) and ii). With the notations of case ii) in Proposition 3.1, Xm is in particular uniquely characterized by
f′(Xm) = 0 with Xm∈ 2θ
5 , θ
, (3.6)
and then we get (3.3).
Step 2: Proof of iii)
In order to compute the derivative with respect to (θ, L), we write the dependance off on (θ, L) as: f(X) =f(X, θ, L). Using (3.6) we have
∂XX2 f(Xm, θ, L) = 3
4(θ−Xm)−5/2−2L= L(5Xm−2θ) θ−Xm
>0. (3.7) Using (3.7), we have ∂XXf(Xm, θ, L) 6= 0. Then using the Implicit Function Theorem, we deduce that Xm = Xm(θ, L) is a smooth function. Now using the definition of Xm = Xm(θ, L), we get
d
dθ(∂Xf(Xm, θ, L)) =∂XX2 f(Xm, θ, L)∂θXm+∂Xθ2 f(Xm, θ, L) = 0. (3.8) Using (3.8) and (3.7), we get (3.4). In a similar way, we get (3.5).
In what follows, we consider the minimizer off on the subintervalD of[0, θ)where we recall that D:=h
0,θe+i
and θe+ is defined in (1.7). For this purpose, we introduce X := minn
θe+, Xm
o
, (3.9)
where Xm is the quantity introduced in Proposition 3.1. Then we have argmin
D
f ⊂ {0, X}. (3.10)
For this reason, we have to study in particular the equalities
Xm = θ;e f(0) = f(Xm);
f(0) = f(θ).e And then we need to consider the following functions
Ld(θ) := 25 24
r5
3θ−5/2 for θ >0; (3.11)
L01(θ) := 55/2
16 θ−5/2 for θ >0; (3.12)
where Ld and L01 have already been introduced in (3.2) and (1.13).
First of all, we have to give some geometrical results concerning the position of such curves describing our domains. For an illustration of the following lemma, we refer the reader to Figure 3.
Lemma 3.2 (Positions of some curves)
We recall θ∗ given in (1.12). The following results hold true:
i) L01(θ∗) =L02(θ∗) = L12(θ∗).
ii) For all θ > θ∗, we have L01(θ)< L02(θ)< L12(θ).
iii) For all θ >0, we have L01(θ)> Ld(θ).
We skip the proof of Lemma 3.2 since it is easy to check the result by simple computations.
Figure 3: Schematic phase diagram for parameters (θ, L)
Lemma 3.3 (Properties of the function Xm−θ)e
i) For θ >0 andL > Ld(θ), we have ∂θ(Xm−θ)e >0, where Xm =Xm(θ, L)andθe=θ(θ, L)e are respectively introduced in Proposition 3.1 and (1.7).
ii) For θ >0 and L=L01(θ), we have L > Ld(θ) and
Xm <θe for 0< θ < θ∗; Xm =θe for θ =θ∗; Xm >θe for θ > θ∗.
iii) For θ≥θ∗ and L=L12(θ), we have L > Ld(θ) and Xm =θ.e Proof of Lemma 3.3
Proof of i)Letθ > 0and L > Ld(θ). According to Proposition 3.1 ii),f admits a non zero minimizer Xm ∈(2θ/5, θ). Using (3.4) and the definition of θein (1.7), we get
∂θ(Xm−θ) =e 3/4(θ−Xm)−5/2
3/4(θ−Xm)−5/2−2L−1 = 2L
3/4(θ−Xm)−5/2−2L >0.
Proof of ii) Let θ > 0 and L = L01(θ). Using Lemma 3.2, we deduce that L = L01(θ) >
Ld(θ) for all θ > 0. Then according to Proposition 3.1 ii), f admits a non zero minimizer Xm ∈(2θ/5, θ). Using (3.6) we have
1
2(θ−Xm)−3/2 = 2L01(θ)Xm with 2
5θ < Xm < θ.
LetYm :=Xm/θ, then we have 1
2(1−Ym)−3/2 = 55/2
8 Ym with 2
5 < Ym <1. (3.13) The uniqueness ofYm shows thatYm is a constant independent ofθ. It is easy to check that Ym = 4/5 is the solution of (3.13). Then
Xm = 4
5θ for all θ >0 and L=L01(θ). (3.14) Using (3.14), we get that for θ >0 and L=L01(θ)
Xm−θe= 4θ
5 −eθ(θ, L01(θ)) = θ 5
44α2 54 θ4−1
, which vanishes for θ =θ∗ and then we get the result.
Proof of iii)Letθ ≥θ∗ andL=L12(θ). Using Lemma 3.2, we conclude thatL=L12(θ)≥ L01(θ)> Ld(θ)for allθ ≥θ∗. Then using Proposition 3.1 ii),f admits a non zero minimizer Xm ∈(2θ/5, θ). A direct computation of f′(θ)e shows that
f′(θ) =e Q(θ, L)
2α3L with Q(θ, L) :=L4−4α3θL2+ 4α5.
It is easy to check that in particular for θ ≥ θ∗ and L = L12(θ), we have Q(θ, L12(θ)) = 0 and then f′(θ) = 0. Using Lemma 3.3 ii), we know thate
eθ=Xm > 2
5θ at the point (θ, L) = (θ∗, L∗) =P. (3.15) Moreover for θ > θ∗ and L=L12(θ), we have
d dθ
θ(θ, Le 12(θ))− 2 5θ
= 3
5 + 2 α2
L312(θ)L′12(θ)>0.
Therefore we deduce that θ >e 2θ/5 for θ ≥ θ∗ and L = L12(θ). Then we conclude that Xm =θefor θ≥θ∗ and L=L12(θ).
Lemma 3.4 (Properties of f(Xm) and f(θ))e
Let f(X) =f(X, θ, L) be given in (1.5).
A) Properties of f(Xm)
For θ >0 with L > Ld(θ) and Xm =Xm(θ, L) introduced in Proposition 3.1, we have A.i) d
dLf(Xm(θ, L), θ, L)<0, A.ii) d
dθf(Xm(θ, L), θ, L)<0.
B) Properties of f(θ)e
B.i) Let (θ, L)∈(0,+∞)2 such that eθ=θ(θ, L)e >0. Then we have d dθf
θ(θ, L), θ, Le
<0.
B.ii) For θ > θ∗ and L > L02(θ), we have eθ >0 and d dLf
θ(θ, L), θ, Le
<0.
Proof of Lemma 3.4
A) Let θ > 0 and L > Ld(θ). Using Proposition 3.1 ii), f admits a non zero minimizer Xm ∈(2θ/5, θ).
Proof of A.i) We have d
dLf(Xm(θ, L), θ, L) =∂Lf(Xm, θ, L) +∂Xf(Xm, θ, L)∂LXm =−Xm2 <0, where we have used (1.5) and (3.6) to get the last equality.
Proof of A.ii) We have d
dθf(Xm(θ, L), θ, L) = ∂θf(Xm, θ, L) +∂Xf(Xm, θ, L)∂θXm =−1
2(θ−Xm)−3/2 <0, where again we have used (1.5) and (3.6) to get the last equality.
B) We recall that θe=θ(θ, L)e is given in (1.7).
Proof of B.i) Let(θ, L)∈(0,+∞)2 such that θe=θ(θ, L)e >0. We have d
dθf
θ(θ, L), θ, Le
= d dθ
L
1 α −θe2
=−2Lθ <e 0.
Proof of B.ii) It is easy to check that θ(θ, Le 02(θ)) = α−1/2 >0 for θ ≥θ∗. Since ∂Leθ >0, we deduce that
θ > αe −1/2 >0 for θ > θ∗ and L > L02(θ). (3.16) Now because θ >e 0, a direct computation shows that for θ > θ∗ and L > L02(θ) we have
d dLf
θ(θ, L), θ, Le
= d dL
L
1 α −θe2
= 1
α −θe2−4α2 θe L2. Using (3.16) we get the result.
3.2 Classification of global minimizers of E
In this subsection, we prove Theorem 1.7.
Proof of Theorem 1.7
Using Theorem 1.3, a minimizer ζ = (ζ1, ζ2) of the energy E is always defined as in (1.8) and (1.9). So we have to identify the value of K ∈ D solving Problem (3.1) in each case.
Case A: (θ, L)∈D0
Case A.i): θ >0 and L≤Ld(θ)
Using Proposition 3.1 i) we deduce that argmin
D
f ={0}. Case A.ii): θ >0and Ld(θ)< L < L01(θ)
Using Proposition 3.1 ii) we deduce that f admits a non zero minimizer Xm ∈ (2θ/5, θ).
Using (3.14), a simple computation leads us to the following
f(Xm) = 0 =f(0) for θ >0 and L=L01(θ). (3.17)
Using Lemma 3.4 A.i), we deduce thatf(Xm)>0 = f(0) forθ >0andLd(θ)< L < L01(θ).
Then
argmin
D
f ={0}. Case A.iii): θ > θ∗ and L01(θ)≤L < L02(θ)
Using Lemma 3.2, we have L ≥ L01(θ) > Ld(θ). Now using Proposition 3.1 ii), we deduce that f admits a non zero minimizer Xm ∈(2θ/5, θ). Using Lemma 3.3 ii), we conclude that Xm >θe for θ > θ∗ and L=L01(θ). (3.18) Using Lemma 3.3 i), we deduce thatXm >eθ forθ > θ∗ andL01(θ)≤L < L02(θ). Therefore argmin
D
f ∈n 0,θeo
. Now, we distinguish two cases:
If eθ≤0, then D={0} and argmin
D
f ={0}.
If eθ >0, we proceed as follows: A direct computation shows that
f(θ) = 0 =e f(0) for θ > θ∗ and L=L02(θ). (3.19) According to Lemma 3.4 B.i), we deduce that f(θ)e >0 = f(0) forθ > θ∗ and L01(θ)≤L <
L02(θ). Therefore argmin
D
f ={0}. This shows that in case A.iii), we have
argmin
D
f ={0}. Case B: (θ, L)∈D1
According to Lemma 3.2, we have L > L01(θ)> Ld(θ) for 0< θ ≤ θ∗. Using again Lemma 3.2, we also have L > L12(θ) > L01(θ) > Ld(θ) for θ > θ∗. Therefore, we have L > Ld(θ) for all (θ, L) ∈ D1. Then using Proposition 3.1 ii), we deduce that f admits a non zero minimizer Xm ∈(2θ/5, θ). Using Lemma 3.3 ii), we conclude that
Xm<θe for 0< θ < θ∗ and L=L01(θ). (3.20) Moreover using Lemma 3.3 iii), we have
Xm =θe for θ ≥θ∗ and L=L12(θ). (3.21) Using Lemma 3.3 i), we deduce thatXm <θefor(θ, L)∈D1. Thereforeargmin
D
f ∈ {0, Xm}. Using (3.17) and Lemma 3.4 A.ii), we get thatf(Xm)<0 =f(0) for (θ, L)∈D1. Therefore
argmin
D
f ={Xm}. Case C: (θ, L)∈D2
According to Lemma 3.2, we have L > L02(θ) > L01(θ) > Ld(θ) for θ > θ∗. Then using Proposition 3.1 ii), we deduce thatf admits a non zero minimizerXm ∈(2θ/5, θ).
i) Forθ > θ∗ and L∗ < L < L12(θ), using (3.21) and Lemma 3.3 i), we deduce thatXm >θ.e
ii)Forθ > θ∗ andL02(θ)< L≤L∗, using (3.18) and Lemma 3.3 i), we deduce thatXm >θ.e Therefore for (θ, L)∈D2, we have argmin
D
f ∈n 0,θeo
.
On the other hand, using Lemma 3.4 B.ii) for θ > θ∗ and L > L02(θ), we haveθ >e 0. Using (3.19) and Lemma 3.4 B.ii), we deduce that f(θ)e <0 = f(0) for(θ, L)∈D2. Therefore
argmin
D
f ={θe}. Case D: (θ, L)∈Γ01
It is easy to check that
argmin
D
f ={0, X} with 0< X =Xm <θ.e Case E: (θ, L)∈Γ02
It is easy to verify that argmin
D
f ={0, X} with 0< X =eθ < Xm. Case F: (θ, L)∈Γ12
Similarly, we can show that argmin
D
f ={X} with X =Xm =θ >e 0.
Case G: (θ, L) =P = (θ∗, L∗) Finally, we can check that
argmin
D
f ={0, X} with 0< X =Xm =θ.e Conclusion: So we have proved that
argmin
D
f =
{0} if (θ, L)∈D0; {Xm} if (θ, L)∈D1; {θe} if (θ, L)∈D2∪Γ12;
(3.22)
and
argmin
D
f ∈
{0, Xm} if (θ, L)∈Γ01;
{0,θe} if (θ, L)∈Γ02∪ {P}.
(3.23) Now using Theorem 1.3, a minimizer ζ = (ζ1, ζ2) of the energy E is defined as in (1.8) and (1.9). Moreover, using (1.11) we get T < L if (θ, L) ∈ D1, and T =L if (θ, L) ∈ D2 ∪Γ12 which shows (1.24). Similarly, we get (1.25) for (θ, L)∈Γ01 or (θ, L)∈Γ02∪ {P}.
3.3 Global minimizers of E and blister’s properties
In this subsection, we prove Theorem 1.3 ii) and Proposition 1.8.
Proof of Theorem 1.3 ii)
a) Case (θ, L)∈D0∪D1∪D2 ∪Γ12
Using (3.22), there exits a unique minimizerK ∈ Dof the function f. Now using Theorems 1.2 and 1.3 i), there exists a global minimizerζ ∈Y of the energyEand there exitsT ∈[0, L]
such that (up to addition of constants and translation of (ζ1, ζ2)) this minimizer is given by (1.8) and (1.9).
b) Case (θ, L)∈Γ01∪Γ02∪ {P}
Using (3.23), there exit exactly two minimizers of the function f on D. Similarly using (1.10), Theorems 1.2 and 1.3 i), we see that the energyE has exactly two global minimizers inY: the trivial solution(ζ1, ζ2) = (0,0) and a blisterζ ∈Y given by (1.8)-(1.9).
Proof of Proposition 1.8
Proof of i)
Let (θ, L) ∈ D1 ∪D2 ∪Γ12. Using (3.22), we conclude that there exists a unique K = K(θ, L)∈ D (see (3.22)) such that
minX∈Df(X) = f(K),
with f and D, respectively introduced in (1.5) and (1.7). According to (1.9), the length of the support of ζ2 and its amplitudeA can be written (using (1.6) to express(θ, L)in terms of (θ, L)) as
T = 2π
s 2α
3(θ−αK) = 2π r2
3(θ−K)−1/2, (3.24)
and
A= s
KL πβ =
2 3
1/4
(πα)−1/2(KLT)1/2. (3.25) Using (3.22), we have
K =K(θ, L) =
Xm(θ, L) if (θ, L)∈D1; θ(θ, L)e if (θ, L)∈D2∪Γ12.
The function θeis smooth. Moreover, Xm is smooth on D1 according to Proposition 3.1 iii).
ThereforeK =K(θ, L) is a smooth function of (θ, L) on each domain D1 and D2.
Straightforward calculations show that for (θ, L)∈D1∪D2
∂θT =π r2
3(θ−K)−3/2(∂θK−1),
∂LT =π r2
3(θ−K)−3/2∂LK,
∂θA = 1
24 1/4
L απKT
1/2
(T ∂θK +K∂θT),
∂LA = 1
24 1/4
1 απKT L
1/2
(KT +T L∂LK +KL∂LT).
(3.26)
Case 1: (θ, L)∈D1
Using (3.4), (3.5) and (3.26), we conclude that ∂θT, ∂LT, ∂θA, ∂LA≥0.
Case 2: (θ, L)∈D2∪Γ12
Using (3.22), we have K =θ(L) =e θ−α2/L2 and T =L. So we have
∂θK = 1 and ∂LK = 2α2 L3 >0.
Using (3.26), we conclude that ∂θT, ∂LT, ∂θA, ∂LA≥0.
Proof of ii)
Step 1: Proof of (1.26)
Our goal is to compute the derivative of T with respect to θ along the curve Γ01. Using (3.26), (3.4) and (3.5), we get with obvious notation for (θ, L) ∈ Γ01 (using the fact that L=L01(θ) given in (1.13))
d
dθT(θ, L01(θ)) = π r2
3(θ−K)−3/2[(∂θK−1) + (∂LK)L′01(θ)]
= π r2
3(θ−K)−3/2
55/2θ−5/2(2−5θ−1K) 16 (3/4(θ−K)−5/2−2L)
.
Using (3.6) and (3.7), we conclude that T = T(θ, L01(θ))is decreasing in θ along the curve Γ01. Then we deduce that inf
(θ,L)∈Γ01
T = T(P) = T(θ∗, L01(θ∗)). Using (1.25), we have T(P) = L∗, whereL∗ is the value of L at the point P. Therefore
(θ,L)∈Γinf 01
T =L∗ = 2π r2
3
L01(θ∗)
α = 4π
r2
3α1/4 =:T∗. (3.27) Using the monotonicity of T in θ and Lon D1, we get (1.26).
Step 2: Proof of (1.27)
LetK :=K/θ. Similarly using (3.25) and (3.24), we explicit Ain term of K for(θ, L)∈Γ01
(in particular L=L01(θ)). A straightforward computation gives A(θ, L01(θ)) =
1 12
1/2
55/4 K(1−K)−1/21/2
α−1/2θ−1.
For (θ, L)∈ Γ01, with K =Xm we have by (3.14) that K = 4/5. Then A= A(θ, L01(θ)) is decreasing inθ along the curve Γ01. So
(θ,L)∈Γinf 01A =A(θ∗, L01(θ∗)) = 2 5
9 1/4
K(1−K)−1/21/2
= 4/√
3 =:A∗. Finally using the monotonicity of A in θ and L on D1, we get (1.27).
Acknowledgment
The authors thank A. El Doussouki for helpful discussions on the problem. This work has been financially supported by the project CEDRE 11EF45 L20 (2012-2013).
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